Prove that the expression `((3n)!)/(3!)^n` is an integer for all `n leq 0`.

Show that the expression (n^5)/(5) + (n^3)/(3) + (7n)/(15) is a positive integer for all n in N .

By mathematical induction prove that, (n^(7))/(7)+(n^(5))/(5)+(2n^(3))/(3)-(n)/(105) is an integer for all ninNN .

Prove by induction that the sum S_(n)=n^(3)+2n^(2)+5n+3 is divisible by 3 for all n in N.

Prove the following by the principle of mathematical induction: (n^(7))/(7)+(n^(5))/(5)+(n^(3))/(3)+(n^(2))/(2)-(37)/(210)n is a positive integer for all n in N.

Prove that (n!)^2 le n^n. (n!)<(2n)! for all positive integers n.

Prove that ((2n)!)/(2^(2n)(n!)^(2))<=(1)/(sqrt(3n+1)) for all n in N

Prove that (n!)^(2)

Prove the following by the principle of mathematical induction: (n^7)/7+(n^5)/5+(n^3)/3+2(n^3)/3-n/105 is a positive integer for all n in Ndot

Prove by the principle of mathematical induction that for all n in N. (2^(3n)-1) is divisible by 7 for all n in N .